Problem: Let $\triangle ABC$ have side lengths $AB=13$, $AC=14$, and $BC=15$.  There are two circles located inside $\angle BAC$ which are tangent to rays $\overline{AB}$, $\overline{AC}$, and segment $\overline{BC}$.  Compute the distance between the centers of these two circles.
Solution: The two circles described in the problem are shown in the diagram.  The circle located inside $\triangle ABC$ is called the incircle; following convention we will label its center $I$.  The other circle is known as an excircle, and we label its center $E$.  To begin, we may compute the area of triangle $ABC$ using Heron's formula.  The side lengths of triangle $\triangle ABC$ are $a=15$, $b=14$, and $c=13$, while the semiperimeter is $s=\frac{1}{2}(a+b+c)=21$, so its area is \[ K = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21\cdot 6\cdot 7\cdot 8} = 84. \]We find the inradius $r$ of $\triangle ABC$ by using the fact that $K=rs$, so $84=21r$, giving $r=4$.  Next label the points of tangency of the incircle and excircle with ray $\overline{AC}$ as $S$ and $T$, as shown at right.  It is a standard fact that $AS=s-a=6$ and $AT=s=21$. (The reader should confirm this.  Repeatedly use the fact that tangents from a point to a circle have the same length.)  Furthermore, the angle bisector of $\angle A$ passes through $I$ and $E$, and the radii $\overline{SI}$ and $\overline{TE}$ are perpendicular to $\overline{AC}$, so triangles $\triangle ASI$ and $\triangle ATE$ are similar right triangles.  By the Pythagorean Theorem we compute \[ AI = \sqrt{(AS)^2+(SI)^2} = \sqrt{36+16}=2\sqrt{13}. \]Using the similar triangles we find that $AI/AE = AS/AT = 6/21 = 2/7$.  Therefore $AE=7\sqrt{13}$ and we conclude that $IE=AE-AI=\boxed{5\sqrt{13}}$.

[asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
draw((0,0)--(4,0)--(3,5)--cycle);
draw(incircle((0,0),(4,0),(3,5)));
real x = 1.15;
pair A = (0,0) + x*(-3,-5);
pair B = (4,0) + x*(1,-5);
draw(A--(3,5)--B--cycle);
draw(incircle(A,(3,5),B));
label("$A$",(3,5),N);
label("$B$",(4,0),E);
label("$C$",(0,0),W);
pair I = incenter((0,0),(3,5),(4,0));
pair iFoot = foot(I,(0,0),(3,5));
label("$S$",iFoot,W);
label("$I$",I,E);
draw(iFoot--I);
pair I2 = incenter(A,(3,5),B);
pair iFoot2 = foot(I2,(0,0),(3,5));
label("$T$",iFoot2,W);
label("$E$",I2,S);
draw(iFoot2--I2);
draw((3,5)--(I2));
[/asy]